We are interested in proposing the convex fuzzy functions from R into E to describe the concepts of left and right Generalized derivatives. In this way, several properties of the concepts are discussed. As well, the notions of Generalized subdifferential for these functions are studied and calculated. Finally, the properties of the concepts are illustrated and applied through some examples.

In this paper, we deal with the subdifferential concept on Hadamard spaces. Flat Hadamard spaces are characterized, and necessary and sufficient conditions are presented to prove that the subdifferential set in Hadamard spaces is nonempty. Proximal subdifferential in Hadamard spaces is addressed and some basic properties are highlighted. Finally, a density theorem for subdifferential set is established.

‎In this paper‎, ‎we extend the notion of approximate convexity to set-valued maps and obtain some relations between approximate convexity and approximate monotonicity of their normal subdifferential‎.

In this paper, we investigate relation between weak subdifferential and augmented normal cone. We define augmented normal cone via weak subdifferential and vice versa. The necessary conditions for the global maximum are also stated. We produce preliminary properties of augmented normal cones and discuss them via the distance function. Then we obtain the augmented normal cone for the indicator function. Relation between weak subifferential and augmented normal cone and epigraph is also explored. We also obtain optimality conditions via weak subdifferential and augmented normal cone. Finally, we define the Stampacchia and Minty solution via weak subdifferential and investigate the relation between Stampacchia and Minty solution and the minimal point.

The purpose of this paper is to develop nonsmooth optimization problems (P) in which all emerging functions are assumed to be real-valued quasiconvex functions that are defined on a finite-dimensional Euclidean space. First, we introduce two linear optimization problems with the same optimal value of the considered problem. Then, we introduce a real-valued non-negative gap function for (P), and we provide some conditions which ensure that its null points are the same as the optimal solution of problem (P). The results are based on incident subdifferential, which is an important concept in the analysis of quasiconvex functions.

In this paper we study optimization problems with infinite many inequality constraints on a Banach space where the objective function and the binding constraints are locally Lipschitz.Necessary optimality conditions and regularity conditions are given. Our approach are based on the Michel-Penot subdifferential.

This article is concerned with non-smooth/nonconvex robust multi-objective optimization problems involving uncertain inequality and equality constraints. Employing some advanced tools of variational analysis such as the approximate extremal principle and the weak fuzzy sum rule for the Frechet subdifferential, we first drive a fuzzy necessary optimality condition of a non-smooth/nonconvex multi-objective optimization problem without any constrained qualification in the sense of the Frechet subdifferential. Then by exploiting the obtained fuzzy optimality condition, the non-smooth version of Fermat’, s rule and formulae for the limiting subdifferential of an infinite family of non-smooth functions, we establish a necessary optimality condition in terms of the limiting subdifferential for weakly robust efficient solutions of the reference problem. Further, we present an example to illustrate this condition for an uncertain multi-objective optimization problem involving equality and inequality constraints. Finally sufficient conditions for weakly robust efficient solutions and robust efficient solutions of the problems are provided by presenting new concepts of Generalized convexity.

In this research, in order to investigate the effect of the piezoelectric patch which is used as a sensor or actuator in rotating flexible structures such as a helicopter blade, the free vibrations of the rotating rectangular sheet with and without the piezoelectric patch have been presented. First-order shear deformation theory is considered for plate displacement and piezoelectric field. Considering the effect of Coriolis acceleration, centrifugal acceleration and centrifugal in-plane forces, the equations of motion are derived from Hamilton's principle and the electromechanical couple equation is obtained from Maxwell's equation. For piezoelectric, two electrical conditions, open circuit and closed circuit, which are used in sensors and actuators, respectively, have been considered. The equations are discretized with the help of the numerical method of Generalized differential squares and the matrices of inertia mass, eccentricity, Coriolis and stiffness matrix are obtained. Natural frequency values for beam and rotating plate have been compared in Abaqus software. Also, the values obtained from the numerical solution in MATLAB have been verified with articles and ABAQUS, which have high accuracy. The effect of parameters such as hub radius, rotation speed, sheet thickness, aspect ratio, piezoelectric patch thickness and applied voltage on the natural frequency of the system has also been investigated.

This paper aims to establish first-order necessary optimality conditions for non-smooth multi-objective Generalized semi-infinite programming problems‎. ‎These problems involve inequality constraints whose index set depends on the decision vector‎, ‎and all emerging functions are assumed to be locally Lipschitz‎. ‎We introduce a new constraint qualification for these problems‎. ‎Building upon this qualification‎, ‎we derive an upper estimate for the Clarke sub-differential of the value function of the problem‎. ‎Furthermore‎, ‎we demonstrate the necessary optimality conditions for properly efficient solutions to the problem‎.